Need help. ASAP!

Question

A. Maria performed the synthetic division below for (x⁴ + 3x² - 2x) ÷ (x - 4). There are three errors. Explain the errors and enter the correct solution:

 

B. Julie applied Pascal's triangle to the binomial expansion of (x - 4)⁵ and her solution is shown below. Explain the error and enter the correct solution:

x⁵ + 20x⁴+ 160x³ + 640x² + 1280x + 1024 = 0

Answer 1

For part A, the errors are as follows,

1. The divisor is (x-4), meaning that 4 is a zero of the function, not -4. So the divisor in the synthetic division should be 4 instead of -4.
2. The first column of the synthetic division in this case is the x⁴ column since we are dividing a polynomial of degree 4 (x⁴ + 3x² - 2x) .Therefore the first term of the answer should be the x³ term, not the x term.
3. There should be another column here for the x³ column between the first and second columns since the polynomial could be written as x⁴ + 0x³+3x² - 2x.

If we correct each of these mistakes, we find the following solution,

x³ + 4x²+19x + 74/(x-4)

For part B, remember that when using pascals triangle to find the binomial expansion, the + or - sign within the parenthesis will determine the + and - signs of the expanded form.

Case 1) We have the form (a + b)ⁿ, then all of the terms in the expanded form are positive.

Case 2) We have the form (a - b)ⁿ, then the sign of the terms in the expanded form alternate from + to -.

For example, in this problem, we have (x - 4)⁵, therefore the signs of the expanded form should alternate.

Julie has written x⁵ + 20x⁴+ 160x³ + 640x² + 1280x + 1024 = 0

but should have written x⁵ - 20x⁴+ 160x³ - 640x² + 1280x - 1024 = 0